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Beliebt Rechnen >

integral of 1/(usqrt(1+u^2))

Lösung

\int \frac{1}{u 1 + u ^{2}} d u

Lösung

- \frac{1}{2} ln 1 + u^{2} + 1 + \frac{1}{2} ln 1 + u^{2} - 1 + C

Schritte zur Lösung

\int \frac{1}{u 1 + u ^{2}} d u

Wende U-Substitution an

= \int \frac{1}{v ^{2} - 1} d v

Faktorisiere

v^{2} - 1 : - (- v^{2} + 1)

= \int \frac{1}{- ( - v ^{2} + 1 )} d v

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \int \frac{1}{- v ^{2} + 1} d v

Nutze das gemeinsame Integral :

\int \frac{1}{- v ^{2} + 1} d v = \frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2}

= - (\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2})

Setze in

v = 1 + u^{2}

ein

= - (\frac{ln 1 + u ^{2} + 1}{2} - \frac{ln 1 + u ^{2} - 1}{2})

Vereinfache

- (\frac{ln 1 + u ^{2} + 1}{2} - \frac{ln 1 + u ^{2} - 1}{2}) : - \frac{1}{2} ln 1 + u^{2} + 1 + \frac{1}{2} ln 1 + u^{2} - 1

= - \frac{1}{2} ln 1 + u^{2} + 1 + \frac{1}{2} ln 1 + u^{2} - 1

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{2} ln 1 + u^{2} + 1 + \frac{1}{2} ln 1 + u^{2} - 1 + C

Graph

Beliebte Beispiele

derivative of 7sin(337x+(7pi/(18)))

\frac{d}{d x} (7 sin (337 x + \frac{7 π}{18}))

integral of (2x+6)/(x^2+2x+5)

\int \frac{2 x + 6}{x ^{2} + 2 x + 5} d x

partialfraction (x^2)/(1-x^2)

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t a y l or 5^{t}, 0